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The widespread use of computer hardware and software in society has led to the emergence of a type of criminal conduct known as cybercrime, which has become a major worldwide concern in the 21st century spanning multiple domains. As a result, in the present setting, academics and practitioners are showing a great deal of interest in conducting research on cybercrime. In this work, a fractional-order model was replaced by involving three sorts of human populations: online computer users, hackers, and cyber security professionals, in order to examine the online computer user-hacker system. The existence, uniqueness and boundedness were studied. To support our theoretical conclusions, a numerical analysis of the influence of the various logical parameters was conducted and we derived the necessary conditions for the different equilibrium points to be locally stable. We examined the effects of the fear level and refuge factor on the equilibrium densities of prey and predators in order to explore and understand the dynamics of the system in a better way. Using some special circumstances, the model was examined. Our theoretical findings and logical parameters were validated through a numerical analysis utilizing the generalized Adams-Bashforth-Moulton technique. [ABSTRACT FROM AUTHOR]

Rumor spreading is the circulation of doubtful messages on the social network. Fact retrieving process that aims at preventing the spread of the rumor, appears to have a significant global impact. In this research, we have investigated a mathematical model projecting rumor spread by considering six groups of individuals namely ignorant, exposed, intentional rumor spreader, unintentional rumor spreader, stifler, and fact retriever. To represent the current abnormal and fast pattern of the message spread around various platforms, in the projected model, we have implemented the fractional derivative in the Caputo context. Using the existing theory of the fractional derivative, we have examined the theoretical aspects such as the existence and uniqueness of the solutions, the existence and stability of the rumor-free and rumor equilibrium points, and the global stability of the rumor-free equilibrium point. Computing basic reproduction numbers, we have analyzed the existence and stability of points of equilibrium. The sensitivity of basic reproduction numbers is also examined. Importance of the fact retrieving drive is highlighted by relating it to the basic reproduction number. Finally, by applying the Adams-Bashforth-Moulton method, we have presented the numerical results by capturing the profile of each of the groups under the influence of fractional derivative and investigated the impact of rumor verification rate and contact rate in controlling and preventing the rumor. With the Caputo fractional operator in the projected model, the current research highlights the significance of the fact retriever and the curb in individual contact and captures the relevant consequences. [ABSTRACT FROM AUTHOR]

In the marine ecosystem, phytoplankton plays a vital role as the primary supplier of oxygen, contributing to 50% of the total oxygen production. Not only does it serve as a significant source of food for other species, but it also sustains life in the ocean. However, the rising ocean temperatures caused by global warming have severely hindered the ability of phytoplankton to generate oxygen. Furthermore, fishes are crucial consumers of oxygen within the marine ecosystem. This research paper presents a model that intricately connects the dynamics of oxygen, phytoplankton, zooplankton, and fish using the Caputo fractional derivative. The model aims to examine the impact of global warming on the collective dynamics by considering the relationship between the rate of oxygen generation, temperature, and time. The paper establishes the existence and uniqueness of solutions and also analyzes the stability of equilibrium points. Numerical simulations are conducted to demonstrate the impact of fractional derivatives and global warming on oxygen depletion and species extinction. [ABSTRACT FROM AUTHOR]

In this paper, we have projected the theoretical and numerical investigation of the mathematical model representing the yellow fever virus transmission from infected mosquitoes to humans or vise-versa through mosquito bites in the framework of the Caputo derivative. Theoretical aspects of the dynamics of susceptible individuals, exposed individuals, infected individuals, toxic infected individuals, recovered and immune individuals, and susceptible mosquitoes and infected mosquitoes have been analyzed by using the theory of fractional calculus such as boundedness, uniqueness and existence of the solutions. Sufficient conditions for the global stability of the virus-free point of equilibrium are inspected. T validate the theoretical results numerical analysis is performed using the generalized Adams-Bashforth-Moultan method. [ABSTRACT FROM AUTHOR]

We suggest a fractional differential equation describing the intake of illegal drugs in a population made up of non-users and drug consumers. Drug users are categorized into experimental users, recreational users, and addicts. This is an effort by analogy with the traditional multi-species logistic Holling type-III predator-prey models to suggest a model that considers non-users as prey, experimental and recreational users as predators as well as preys and addicts as predators. The proposed model is analyzed in terms of boundedness, existence, and uniqueness of solutions. We derive sufficient conditions for existence and stability of points of equilibrium in terms of some threshold parameters. The dynamics of these threshold parameters and the sensitivity indices of parameter values are investigated by relating them to basic reproduction numbers. The theoretical results are then validated by numerical simulations. [ABSTRACT FROM AUTHOR]

Wireless sensor networks (WSNs) are subject to cyber attacks. Security of such networks is a significant priority for everyone. Due to the network's frail defense mechanisms, WSNs are easy targets for worm attacks. A single unsecured node via contact can effectively propagate the worm across the entire network. Mathematical epidemic models are helpful in analyzing worm propagation in WSNs. To understand the attacking and spreading dynamics of worms in WSNs, a fractional‐order compartmental epidemic model is investigated with susceptible, exposed, infected, recovered, and vaccinated nodes. Dynamical aspects such as boundedness, existence, and uniqueness of the solutions are presented with the help of fractional calculus theory. Global stability of the points of equilibrium are established. The projected nonlinear structure is examined numerically via the generalized Adams–Bashforth–Moulton method. This study demonstrates the influence of the fractional operator on WSNs dynamics and the efficiency of the numerical method. [ABSTRACT FROM AUTHOR]

In this paper, we have derived a new operational matrix of fractional integration by using the independence polynomial of a complete bipartite graph and applied it to solve the Poisson equation with Dirichlet boundary conditions. While deriving the operational matrix, the Caputo sense fractional derivatives are considered. Series solutions are found by using the collocation matrix method. The main characteristic of the approach is that it reduces a complex fractional differential equation to a system of algebraic equations. The error bound and computational complexity of the projected algorithm are also investigated. Solutions obtained for the Poisson equation have established the relevance and applicability of the method described. Comparative analysis of the obtained results with the exact solutions convinces that the present method can be considered an efficient numerical tool for solving fractional differential equations. [ABSTRACT FROM AUTHOR]

The Korteweg{De Vries (KdV) equation has always provided a venue to study and generalizes diverse physical phenomena. The pivotal aim of the study is to analyze the behaviors of forced KdV equation describing the free surface criti- cal ow over a hole by finding the solution with the help of q-homotopy analysis transform technique (q-HATT). he projected method is elegant amalgamations of q-homotopy analysis scheme and Laplace transform. Three fractional oper-ators are hired in the present study to show their essence in generalizing the models associated with power-law distribution, kernel singular, non-local and non-singular. The fixed-point theorem employed to present the existence and uniqueness for the hired arbitrary-order model and convergence for the solution is derived with Banach space. The projected scheme springs the series solution rapidly towards convergence and it can guarantee the convergence associated with the homotopy parameter. Moreover, for diverse fractional order the physi- cal nature have been captured in plots. The achieved consequences illuminates, the hired solution procedure is reliable and highly methodical in investigating the behaviours of the nonlinear models of both integer and fractional order. [ABSTRACT FROM AUTHOR]

The pivotal aim of the present study is to find the solution for a nonlinear system describing the aggregation process of cellular slime mold by using q -homotopy analysis transform method. The coupled system is considered within the frame of the Caputo fractional operator. We examine three different cases with distinct values of sensitivity function χ ρ = 1 , ρ and ρ 2 to exemplify the efficiency and applicability of the considered scheme. We capture the nature of the obtained results with respect to the fractional order, with distinct initial conditions, and illustrate them using 2D and 3D plots for particular values of the parameters. The considered scheme offers parameters, which help to adjust the convergence region, and we plotted the ℏ-curves to dissipate the effect in the present framework. Moreover, some simulating and important behavior of the considered model using attained results shows the prominence of the hired operator while analyzing the coupled equations and confirms the competence of the projected scheme. [ABSTRACT FROM AUTHOR]

In this study, we depicted the spread of pests in tea plants and their control by biological enemies in the frame of a fractional-order model, and its dynamics are surveyed in terms of boundedness, uniqueness, and the existence of the solutions. To reduce the harm to the tea plant, a harvesting term is introduced into the equation that estimates the growth of tea leaves. We analyzed various points of equilibrium of the projected model and derived the conditions for the stability of these equilibrium points. The complex nature is examined by changing the values of various parameters and fractional derivatives. Numerical computations are conducted to strengthen the theoretical findings. [ABSTRACT FROM AUTHOR]

Investigation of the dynamical behavior related to environmental phenomena has received much attention across a variety of scientific domains. One such phenomenon is global warming. The main causes of global warming, which has detrimental effects on our ecosystem, are mainly excess greenhouse gases and temperature. Looking at the significance of this climatic event, in this study, we have connected the ACT-like model to three climatic components, namely, permafrost thaw, temperature, and greenhouse gases in the form of a Caputo fractional differential equation, and analyzed their dynamics. The theoretical aspects, such as the existence and uniqueness of the obtained solution, are examined. We have derived two different sliding mode controllers to control chaos in this fractional-order system. The influences of these controllers are analyzed in the presence of uncertainties and external disturbances. In this process, we have obtained a new controlled system of equations without and with uncertainties and external disturbances. Global stability of these new systems is also established. All the aspects are examined for commensurate and non-commensurate fractional-order derivatives. To establish that the system is chaotic, we have taken the assistance of the Lyapunov exponent and the bifurcation diagram with respect to the fractional derivative. To perform numerical simulation, we have identified certain values of the parameters where the system exhibits chaotic behavior. Then, the theoretical claims about the influence of the controller on the system are established with the help of numerical simulations.

In order to depict a situation of possible spread of infection from prey to predator, a fractional-order model is developed and its dynamics is surveyed in terms of boundedness, uniqueness, and existence of the solutions. We introduce several threshold parameters to analyze various points of equilibrium of the projected model, and in terms of these threshold parameters, we have derived some conditions for the stability of these equilibrium points. Global stability of axial, predator-extinct, and disease-free equilibrium points are investigated. Novelty of this model is that fractional derivative is incorporated in a system where susceptible predators get the infection from preys while predating as well as from infected predators and both infected preys and predators do not reproduce. The occurrences of transcritical bifurcation for the proposed model are investigated. By finding the basic reproduction number, we have investigated whether the disease will become prevalent in the environment. We have shown that the predation of more number of diseased preys allows us to eliminate the disease from the environment, otherwise the disease would have remained endemic within the prey population. We notice that the fractional-order derivative has a balancing impact and it assists in administering the co-existence among susceptible prey, infected prey, susceptible predator, and infected predator populations. Numerical computations are conducted to strengthen the theoretical findings.